Combine red food coloring and 1/3 cup water in a ramekin or small bowl; do the same with the blue food coloring in a separate ramekin.

Gently insert 3 or 4 toothpicks about 1/4-inch deep through the center of an egg and balance it so that one end is dipped in one of the dyes. (Don't pierce the eggs too much, or they'll break after you halve them). Let stand for 5 to 10 minutes, depending upon how strong you like the colors. Dip the other end of the egg in the other dye so the eggs have red, white, and blue bands. Dye the rest of the eggs and let dry on paper towels.

Slice colored eggs in half lengthwise and set the whites aside on a serving platter. Push the yolks through a sieve into a bowl or use a potato ricer.

Mix mayonnaise, horseradish, rice vinegar, salt, and pepper into the yolks and stir until smooth. Taste and adjust seasoning. Transfer filling into a piping bag fitted with a large star tip.

Pipe filling into each egg white half. Place smoked paprika in a fine mesh strainer; hold the strainer over each egg half and tap gently to dust with paprika. Sprinkle deviled eggs with chives. Chill until ready to serve.